Steady state analysis of a nonlinear renewal equation

Steady state analysis of a nonlinear renewal equation

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Article ID: iaor20113008
Volume: 53
Issue: 7-8
Start Page Number: 1420
End Page Number: 1435
Publication Date: Apr 2011
Journal: Mathematical and Computer Modelling
Authors:
Keywords: linear algebra, renewal processes, system dynamics
Abstract:

In the analysis of a nonlinear renewal equation it is natural to anticipate the existence of nonzero steady states and deal with the question of their stability. Sufficient conditions for existence and uniqueness for these steady states are given. The study of the linearized version of the renewal equation around the steady state helps to a great extent to have insight into some complicated dynamics of the full problem. At this stage the first eigenvalue of the steady state plays a vital role. The characteristic equation, a functional equation whose roots are the eigenvalues, is derived. We give various structures showing that the steady state may be stable or unstable (though the fertility rate is decreasing with competition). A similar study is carried out on a nonlinear model motivated by neuroscience in which the total population is conserved.

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